Most people just use a (large) hard drive in their original xbox and rip it that way but it can be done on a PC.
It is not possible with off the shelf hardware though (although you can flash some drives), you need either a kreon drive (or make one yourself; this is the flashing thing I mentioned) or you can use a 360 drive with 0800 mode (built in with the 1.6 ixtreme line or as a separate firmware prior to it*).
If using the integrated 0800 mode then you will need the tray half open thing and if using the dedicated then it should not matter.
Anyhow that over xbox backup creator is the tool of choice here.
A guide to it:
(ignore stuff on kreon firmware and different, the guide is outdated in that respect as everyone either uses kreon or 0800 modes these days)
You will also want a copy of dvd mulleter (abgx360 replaced it for 360 games but for original xbox stuff it is still good).
The only real trouble with all of this (other than a game clocking a few gigs and still costing a dual layer disc) is when you try to use scene releases or releases that have otherwise seen ripping, hacking or downsizing. This should not apply here though.