This is a fascinating problem because it shows just how tricky it is to define units of measure. In school, we are often given the impression that the units are inviolate structures fundamental to nature, but reality shows otherwise.
This task would have been impossible before 1960. From 1889 until 1960, the meter was actually defined by a particular bar stored in Sèvres, France. Without access to it, it would be virtually impossible to construct a 1 meter object. Before that time, the meter was actually defined to be 1/10,000,000 of a quadrant along the Earth's meridian. Interestingly enough, the meter bar they constructed was actually 200um shorter than it should have been due to a miscalculation, but once the bar was struck, it became the meter.
From 1960 to 1983, the meter was redefined to be 1650763.73 wavelengths of light from a specified transition in krypton-86. This meant that, for the first time, one could have a definitive "meter" that was not bound to a physical object which could be damaged or worn down. This definition was replaced in 1983 with our present one, which is the distance light travels in a vacuum in 1/299,792,458th of a second. This definition locked the meter down to a physical constant (the speed of light), and the second.
Swallow the spider to get to the fly.... okay, how do we measure a second to within nanosecond precisions?
For the longest time, the "second" was measured as a fraction of the day. This was sufficient for centuries. However, the day actually varies slightly, so in 1956 we redefined it to be "the fraction 1/31,556,925.9747 of the tropical year for 1900 January 0 at 12 hours ephemeris time." Yes, this is how we do things in science.
Of course, it would be easy to lose track of exactly how long the period of year was back in 1900, so this was less than ideal. In 1967, after the invention of the atomic clock, it was redefined to be "the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom." What a mouthful!
So these numbers are incredibly precise. They need to be. If you want to measure the second accurate to 1 part per billion (one nanosecond) so that you can measure a meter to 1 part per billion (nanometer), you need all those sig figs. If you miss just 10 of those tiny transitions in your caesium atoms, you're off by a nanosecond!
At this scale, silly things start to matter. For instance, we noticed that time was traveling faster for some clocks at different altitudes because of relativistic effects. Yes, relativistic effects matter on these scales, so the second was re-clarified in 1997 to be the rate of an atomic clock operating at mean sea level! This had about a 0.1ns difference per second, so it shows up on the scales you care about.
So to answer your question, no. A single person is not about to make an accurate meterstick in their lifetime. Your first step would be to create an atomic clock, which calls for high purity components, high vacuums, and quite a lot of high precision machining. With that, you could acquire some caesium-133 and measure a second to a high enough precision. Then you could attempt to measure the speed of light using another expensive scientific instrument to create your meter.
Finally, you could create your meter stick. Did you know that this is not easy? The machinists and metrologists that build these high precision measuring devices are extraordinary. Hopefully your isolated human sent back ten thousand years happens to have spent his whole life mastering this art!
You could try to take a step back in history, and use the krypton definition. You could build an interferometer to do this measurement. However, the transition to measuring the meter based on the speed of light was partially done because the best scientists in the world were having trouble measuring more accurately than about 0.2nm. Shifting to a definition based on the speed of light let them measure frequencies, which were far easier. If the best scientists and metrologists of our world had trouble measuring accurately on the scale you are interested in, its highly likely that you'll have trouble with it 10000BC.
Fortunately, you don't need a kilogram definition. That one is still defined by the IPK in France, a lump of platinum irridium. There's a current effort to change this definition, redefining the kilogram as "1000/27.9769265325 · 6.02214179×10^23 atoms of Si-28. Yes... that qualifies as a "better" definition of the kilogram. Metrology is an insane art, but you have to respect their immaculate precision!