Solving a Wiring Mess?

FueledByRamen asks: "While trying to run a new power line for a large Sun mass-storage cabinet (located nowhere near a 220 outlet of course), I had the misfortune of needing to pop the lid on my main power distribution panel (previously opened in the late 80s). The whole thing is a rats nest and probably a fire hazard - old-style wiring with broken-down cloth/plastic insulation strewn everywhere, and the utility's incoming power cables have some sort of junction in them that's the size of a 1-liter bottle (on each wire) and is covered in layers of electrical tape. Even (gently) putting the panel back on jiggled something important, and there was a nasty cracking noise and half the breakers blew (all breakers in one of the 2 columns). I've worked with mains voltage in the past (wiring new rooms, installing lighting), but nothing on this scale, both in terms of complexity and potential for death. How do you industrious Slashdot readers go about fixing a mess like this (on a tight budget, no less) without getting a mains-induced glimpse at the great beyond?"