Actually what your arrow is pointing at is the pressure tank for the brake system...I would say the pump that pressurizes the brakes is probably what keeps running but I don't know. I'm honestly not sure what you would need to replace, you might try replacing the relay on the firewall (the one with the brown bottom, sticking out from under the plastic shield thing, right above the master cylinder...relay costs about $15 at the parts store). There's a writeup in the faq's section on www.turbotbird.com about diagnosing problems with the ABS system.
The pump or something just plain quit working on mine and I converted over to vac-assisted brakes It's expensive to fix that stuff.
The round cylinder is the ABS accumulator. That's not the problem. The relay is sticking closed so power is continueing to flow to your booster pump. That's what your hearing. Here's the relay diagnosis from NATO.........................................................
Brake Booster Relay Diagnosis - by Jeff Korn
Find the relay with the T/Y, GY, GY/R, GY/Y, PK/LB wires. Remove the relay from its socket, and jumper the T/Y and GY/R terminals together with a heavy gauge jumper wire. Turn the key to run. You should be able to hear the hyd. pump running in the ABS unit. If you hear it run, but don't let it run for more than 10 seconds or so. If you hear it run, turn the key off and put the relay back in its socket. Now, turn the key on, and depress the brake pedal several times and see if the pump runs.
If it doesn't run, probe the PK/LB wire (stick pin thru wire, etc), and, with key on, ground this wire. Pump should run. If it does, the relay is good, if it doesn't run, but ran when the T/Y and GY/R wires were jumpered, then the relay is bad.
The relay is located under this plastic cover (white arrow). I think it's the first one. Also check for a short in the wiring.