I have these shoes that I've worn for a few years, but they're actually still quite okay, if it wasn't for the washed-out color. That makes them look very worn out and battered.
The fabric is linen and they used to be a denim-style dark blue.
I'd really like to give them back some color so I can keep wearing them, but short of painting them all over with a thick marker, I have no idea how this could be achieved with lasting effect. I mean your standard methods for dying fabric (which I don't know what they are) are probably predicated on the fabric not being sown to a rubber sole etc. - right?
Any suggestions would be much appreciated!

If you use a dye that is designed only for natural fibres, it shouldn't affect the sole. The insides will end up blue too though. Be careful of common dyes like the Dylon range, they're designed to work on synthetic materials too, so they'd probably stain the soles.

I've never tried this, but looks fun:

https://www.wikihow.com/Dye-Canvas-Shoes

Looks like your outsoles are darkish rubber, so I wouldn't worry about the dye staining them.

I would remove the laces and insoles first, and you'll want to wash thoroughly after. You probably won't want to wear white socks with them for a while.

You could use a typical natural fiber dye (Rit is the only brand I know, but claims to be all-purpose) or indigo dye. Indigo dying is a bit more involved from what I've heard, but can get much darker and lasts longer.

You should bring them into a mom and pop shoemaker/repair shop if you have one in your area. Shoemakers are generally very wonderful and very crafty people (not just when it comes to shoes). They'll probably be more than happy to give some recommendations for diy if you're strapped for cash, but the amount they charge for their services are usually so low that I'll try go give them at least double.

I'd totally pay a repair shop to do it for me (even though we're on diy here), if I could find one - they're kinda rare. I actually remember buying some shoelaces at a tiny shoe service booth in a mall - but I doubt they would offer things like dying canvas shoes. Will try to call them though and ask, thanks for the tip!

https://duckduckgo.com/?ia=web&origin=funnel_home_website&t=h_&q=Cobbler+near+me&atb=v547-3

This. It’s called “overdye” sometimes.

As a bonus, you can invite your friends to dye stuff at the same time. Do it in a bucket. Got something that’s stained? Dye it!