I just happen to have a drawer full of Compaq Conturae. Neat little machines for their time. Haven't done anything with them for years except run automobile OBD diagnostics every year or two to find out why the Check Engine light is lit THIS time. And even that was superceded a few years ago by a USB OBD2 reader on a netbook. I pulled one of the Compaqs out and find that if has a Linksys EC2T Ethernet card in its' PCMCIA slot. I'm sure that WFWG 3.11 will do ethrnet file transfers given an appropriate driver. Maybe such a card can be found on eBay or some such. I think I recall that I had an early version of Slakware running on one of them at one point. No X windows -- too slow to be useful, but the console was OK.
Another possibility is to remove the hard drive and connect it to another machine. It's been a loooooong time, but ISTR that it's just a bog standard 2.5 inch IDE drive and I think they still make adapters that should work with it.. I seem to recall that access to the drive was not all that difficult, just removing a few screws and separating the case. But It's been 15 years or so and it might be harder than I remember.