Low cost high precision linear position sensor
The only think I know about in that range is the glass scale encoders used in machine tools. They provide a quadrature output (so you can determine direction) but are not absolute sensors, so you do need a way to home or "zero" them in a known position. 

Typical resolution is 5um, so twice as good as your requirement, and claimed accuracy is 5 or 10um typically, depending on length. 

You can find such Chinese scales on eBay and other places in various lengths, starting at two or three hundred dollars. We use them on a milling machine (3-axis) and lathe (2-axis) to speed up manual machining- you have to get the alignment reasonably close when you're installing them, but it's not all that difficult with shims and such like. 

Brand-name ones such as Mitutoyo cost more, of course. This is a tiny fraction of the cost of Heidenhain and similar high-end encoders (I work with such a system that has a resolution measured in nanometers, and with a similarly impressive price tag). 

Here's what the scales look like. The aluminum part moves relative to the slider with the cable attached.