This is a minor problem when writing a fantasy world, but more persistent than you might think. If you want to indicate a character’s height, do you uncritically adopt “feet,” “inches,” “meters”—which may well rub out the sense of this being a different world—or do you invent a measurement of your own—with the proviso that most people won’t understand it, and you’ll need to find a standard of comparison that makes sense? Does your world have a solar calendar? Why not a lunar one? Are the mathematics precise enough to tell the exact length of the year, anyway? And what happens if the length of the year is different from Earth’s?
For all these little problems, this rant has some answers.

1) Don’t be afraid to sacrifice some accuracy.

Say that you’ve chosen the measure “fingerspans” to talk about small lengths, but you can’t find a way to correlate that precisely with inches. So your audience will knowroughly how long this bolt of cloth is, but not that it’s five-and-three-quarters inches.
Does that matter?
Frankly, no. Precision has its place in fantasy, but only with objects and people characters know well, or those they’ve been trained to estimate with an expert eye, especially in the absence of instruments like tape measures that can give an exact number. I highly doubt your character walks down the streets thinking, “I just passed a man six feet and five and three-quarters inches tall…now there’s a woman who has hair precisely eighteen centimeters long…he can cover fifty-five and a half miles a day on a horse like that.” She is likely to know a friend’s height better, and a distance she’s traveled herself is easy enough to estimate. If she’s a dressmaker, she can probably guess how long a bolt of cloth is rapidly, and be close to correct. But if she’s someone picking up the cloth casually in the marketplace, then she can’t know just from one look. She’ll get someone to measure it for her if she really needs an exact length.
And if the precise measurement is not important to the story, why do you need it?
Use the measurements your characters would know and be comfortable with. Points 2, 3, and 4 offer some tricks to suggest the right measurement, but not for reducing it to an exact equivalent with Terran ones. I really don’t think that’s important.

2) Remember that distance and time can march together.

I’ve developed a loathing for adopting miles as a measure of distance lately. However, “leagues” is really no better (especially since that term has different definitions). And saying that it’s five hundred eklas to the nearest town doesn’t help, not without further information. Is that several days’ travel? A month’s? Or just a day, because eklas are measured by how many steps someone takes in a set amount of time?
This is why you use time instead. “This horse could trot for sixteen tuvali, or two days.” That doesn’t reduce each individual tuval to seven-eighths of a mile or fourteen kilometers, but it starts suggesting that the tuvali are fairly large measurements. If the nearest town is a day and a half’s travel away on horseback, or twelve tuvali, that narrows it down still further; you can be fairly confident in correlating four tuvali to a half-day of travel. And there will probably be other clues as the story continues.
Alternatively, you can avoid specific measurements of distance altogether, or say that your world is so bucolic or so cosmopolitan (see point 3) that all sorts of different measurements are used and it’s a bitch trying to track them all and you won’t do it. To hell with it; everyone talks about two days of travel, a half-year of travel, or “as much time as it takes a strong man to get a good sweat going and swing his arms a bit.”

3) Show different measurements from difficult cultures, as a means of comparison.

If one culture uses measurements your readers are familiar with, then you can compare it to the “alien” measurements of the other and show equivalencies that way. (I still wouldn’t recommend detailing exact lengths, times, and distances on every freaking page. The point of different measurements in the first place is to make your fantasy world a little more alien and interesting for your readers, not to penalize them if they don’t know quadratic equations and the precise conversion between Fahrenheit and Celsius).
Say that one culture uses years for a measurement of time. The other uses a measurement called cycles. (Why the difference? See point 5). If one culture is forever talking about “sixteen years” since an important event, and the other culture is forever talking about “sixteen cycles,” you can trust your reader to figure it out. (See point 6). Then you can switch back and forth depending on what character is talking, or leave the familiar measurement behind altogether and venture forth with the other one once it’s sufficiently established.
To return to distance, perhaps one group of people does use leagues to measure the road between cities, but the other uses a length called the strip, based on the number of strips from the skin of a donkey that were originally cut and laid to form the road. (You can have a lot of fun with this. See point 5 again). There’s no reason to pop up and helpfully remind the reader, every time someone thinks Ten strips to the resting place, that “ten strips” means “three leagues.” Believe me, your reader will get this. (Which is why I’m sure I don’t need to remind you of point 6 again).
What about weight? I still think, “Weighs as much as a sack full of stones,” is helpful and illustrative. But if you need precise measurements, then you can, say, conduct a market exchange, in which your hero going up to his friend’s hidden sanctuary is buying enough fresh vegetables to last him for a while in the wilderness. How much do the vegetables weigh? If three carrots, a pod of peas, and six rutabaga are five justai by the merchant’s scale but three pounds according to the measures the character learned and has to translate the justai into, then the reader can get a rough idea of what justai are, and other scenes will help to establish more exact parameters. Weight and length are more easily handled than distance, I think, precisely because they have an easy outlet in commercial scenes, while time can correlate with natural phenomena.

4) Don’t forget the human (or nonhuman) body.

How long is that woman’s hair? To the middle of her back. How thin is that man’s neck? Just thin enough that the protagonist can encircle it with her hands. How long is this bolt of cloth? As long as the distance from my wrist to my shoulder, with a little left over to flow onto my shoulder. How far can that dragon fly without resting? From one end of the mountain range to the other.
You have an understandable, convenient, and reliable source of measurements available to you in your characters’ bodies, and a set that many people use every day. That goes double for things that your character needs to carry or otherwise handle.
No, there’s no reason to name every measurement in your lexicon after some part of the body. But “He was dangling next to a stone wall, three lengths of his body from the ground” is handy as all hell, especially for moments when the character is a bit busy right now, perhaps because he is dangling next to a stone wall, and doesn’t have the time sit around making calculations.

5) Give your measurements a cultural basis.

To return to the case of cycles and years I mentioned above: why do these variations imply about the conception of time? “Cycles” implies a flowing circle of time, the same events repeating over and over again. “Years” probably conjures up a picture of linear time and a number of distinct seasons. Perhaps the people using cycles live near the equator, where seasons do not vary as sharply, while the people using years live in a temperate climate and rely on agriculture. That’s a very basic and simple explanation, but it can become more complicated once you have a basis for it.
Think, too, about events that one culture might consider important and another might not. Most fantasy cultures I’ve read about celebrate their New Year at Midwinter or Midsummer, but that’s by no means universal; the European New Year was once the spring equinox. And what happens in the aforementioned tropical climate, where the change of seasons often means an increase or decrease in the amount of rain, rather than a massive withering or growth? Perhaps the beginning of the new year there is the first day that a certain channel overflows with rainwater, or the day that the first hurricane comes blowing in from the sea. That’s not something a culture in a solidly temperate zone would have to think about. Or the new year might celebrate a holy day or a great cultural hero—something none of the other peoples around them relate to.
Keep in mind that people tell legends about the origin of their cultural artifacts even when those legends aren’t accurate. Thus the old tale that certain English measurements came from the body measurements of kings, which was insisted on long after they were standardized. Perhaps your built culture considers that all their careful measurements of length—thumbspan, fingerspan, span, multispan, and their kindred—as the relics of the massive, more advanced society that occupied the area before them, when in reality they carried them in during their migration from the north.
When you have a cosmopolitan world, knowing the different cultural bases for their measurements will also tell you something important about them, and contribute to your worldbuilding.

6) Trust your readers.

They are not idiots. You can imply the “truth” of your measurements and slowly guide your readers towards them. This is why you don’t have to pop up on every page “helpfully” detailing exactly how many pounds are in fourteen stone or insisting on the strangeness of a lunar calendar with thirteen months, because there are actually thirteen full moons in a year. They can flip back and forth if they get confused. They can work out inferences. If you’re basing the measurement on some archaic measurement in the real world, they can go look it up if they get curious enough.
The worst embarrassment of all is having a glossary in the back of the book that contains all the exact conversion rates. Really? I suppose you can have it if you want, but if your world needs a glossary like that in order to seem whole, that tells me the writing itself is insufficient to create the sense of a new and wondrous place.