What would cause a 1994 Honda Accord's break lights to stay on even when the engine is off?

Check your brake pedal arm where the brake light switch touches it. You'll most likely find a hole where a plastic plug should be and is letting the brake light switch plunger stay extended. An easy fix is to superglue a coin over the hole or simply insert a body panel type clip into it which you can get at your local auto parts store.

The ck eng light will come on every time you turn on the key is the eng turning over when you turn the key to start

This is a matter that you should pay immediate attention to.
The oil light can indicate low oil level in your engine, or low oil pressure. Both can be fatal to your engine.

Check your oil level first.
Look for evidence of leaking oil (possinility a loose oil filter)
The oil pressure sending unit or connections could also be causing the problem

What i advice you is to take out the positive battery terminal and let it rest for about 1 hour,then replace the terminal that should reset it.Do it that worth nothing doing it.

replace head light main switch
or remove fuse,
or put a main disconect to the batery
probably has a short at the tail light asembly sockets , clean the sockets and wires , check the trailer harness.
Vitally

It may be on because the fluid level is low, fill it a little. I should be a DOT3 or DOT4 brake fluid,

is your e-brake on?

check your brake fluid level

Your brake light switch is most likely bad. Under the dash there is a switch that when you push the brake pedal, a pin extends to complete the circuit. There should be a bracket with a rubber tab on it attached to the pedal lever itself. If the rubber tab is missing, it will also allow your brake lights to stay on.

brake in a car is verry important for your saffety and others. that why they install many devices or systems to control them and make shure they work propely 1; brake light switch on the pedal floor(to indicate to the folowing car that you are stoping) 2;brake sensor fluid level (on the brake master cylinder under the hood)
to monitor the level of the fluid in the reservoir 3; brake pressure sensor (on the brake dispencer under hood) will warn you in case of brake pressure faillure .note, it may be more sensors, depending on the make of the vehicle but this is the tree comon causes for brake ligth to stay "on"