It’s not compartment syndrome. A boot won’t help.
Neurology is a good idea. It sounds like a peripheral neuropathy of some sorts. This can sometimes be related to vitamin deficiencies such as B12/folate and occasionally I’ve seen some cases where the person had low iron levels and we corrected those and the symptoms improved. It can be related to diabetes or other neurologic conditions.
Because it’s both sides, it is unlikely to be related to a nerve that is getting pinched in the legs. However, it is possible that this could be related to your low back. I’m not sure if that has been looked at.
As a sports medicine doctor, what you describe sounds less like a sports medicine problem and more like a neurologic issue. Peripheral neuropathy would be the broad term for most likely cause.