Simplest interferometer

Well, one problem you are likely to encounter is that your setup will likely be vibrating with amplitude on this order.. do you have a floating optical table?

Coherence is probably not that important. At a minimum you will need a beamsplitter, two mirrors and a diode or some other way to measure the interference pattern. A lens or two to magnify the pattern will also be helpful.

Lasers aside, the mirror translation stages are likely to be your biggest hassle. If you want to measure sub-optical-wavelength distances, then you should expect to need translation stages that are stable to this sub-micron range or your measurement will wash out. That is, you can have static interference patterns, but if you expect to move mirrors and count fringes to do a length measurement, then your mirror-mover should be stably controllable to the length scales you plan to measure.

On a practical level, this means you will really need a quite solid optical surface like, say, this one, and your mirrors should also be stably mounted with precision screws. If you have a moderate budget then this is nothing to balk at and I expect$^1$ you should really be able to build a quite reasonable one for something like $$100 or $200.

This is not to say, however, that you can't have something really good for really cheap - it just says that you won't be able to finely control the interference pattern. If seeing the pattern is all you really need, then resources like this instructable seem to show that getting a Michelson-Morley interference pattern is relatively easy if you have the patience.

$^1$Note, though, that I'm a theorist.