Please indicate the time period you're interested in for this question. Native Americans live in the present as well.

Since this is a question that covers hundreds of languages and cultures, here's a few different ways that I'm aware of.

In the US, most use miles, in Canada, most use kilometres, though I've noticed that in communities not connected by roads (i.e. most northern communities) miles win out. And time is measured in hours. That's one set.

Okay - I'm Metis, and for us, both time and distance were at one time often measured in pipes - i.e. smoke breaks. portage A is X number of smoke breaks from portage B. Beyond that, we divided time into night and day, morning and evening, weeks, and generally 4 seasons.

I've worked with South Tsimshian. Here, most distances were divided into days/months of travel by canoe. Time was divided into day/night, morning, evening, lunar months, half months, and also into seasons based on both weather (rainy, wet, winter, later rainy) and on food (time of herring, salmon, berries, shoots).

I now work on the Nuxalk language. It is the same as the previous - there's series of numbers specified for counting how many days' travel, and also for counting months of travel. Seasons are linked to both weather, and very closely to the type of fish coming up the river. In addition (I suspect this was true of South Tsimshian as well, but I don't remember), during the dead period of the harvesting cycle, time was divided into types of ceremonies and spiritual/cultural activities that took place.

Another thing to note is that many native american/ First Nations communities had very strong knowledge of any geography they would be talking about. Sguuxs' relationship to geography has been grammaticalized into the language to the extent that there are clitics for locations and specific geographical boundaries (between the forest, the shore, and the water, for example). Nuxalk uses a three-way division of determiners referring to distances within sight/sound of the speaker, beyond sight and sound of the speaker, but still in the Nuxalk valley, and then a third set for distances outside of the valley. So in these situations distances are measured every time a person says the equivalent of "this/that".

Since I'm away from most of my references at the moment, I'll start you out with some an archaeological example rather than a more recent one. Looking way back, during the Middle Woodland period (~100BCE to 500 CE) the Hopewell apparently had a standardized system of linear measurements, which we can now detect in the regularity of of their monuments.

The unit as recognized by archaeologists is known as the Observatory Circle Diameter (OCD). It's roughly equivalent to 1054 feet or 321 meters. It shows up repeatedly in most of the Hopewell's large geometic earthworks. At Newark, it's the diameter of the Observatory Circle, obviously, but it also was used to construct the Octagon that connects to the Observatory Circle. The size of the Square was built so that it's area would match Observatory Circle's area, while the Great Circle's size was built so that it's circumference would match the Square's perimeter. Multiples of the OCD were used to space out the four geometric figures in relation to each other and features of the landscape. Fractions of this unit also show up in some of the smaller features of the site as well. The OCD itself is probably a multiple of one these smaller units that really would have been the primary unit for the Hopewell, but on the scale of monuments the OCD is much more visible today so it current takes precedence.