Here are
some pictures from my installation.
Your '80 MGB has the same instrument panel mine does. The opening will fit most (if not all) standard single-DIN radios. There's no standard for the depth of the radios, and although just about all should fit you should check the measurements on the radios you're looking at. Crutchfield.com is a good source for what's available, and the manufacturer's websites should have the actual radio dimensions on them.
Speakers are another issue. I don't know what's on the market now, but it was tough to find something that fit the "stock" holes in the doors without using big spacers. I've seen everything from simple small spacers to get the speakers out 1/8-1/4" to big wooden boxes that make the speakers look like something from Frankenstein
To clear the window mechanism the door allows 40mm deep speakers (depending on your door panel). You can fudge it 1-2mm by gluing nylon washers to the back of the door panel or door where the mounting screws are. The pre-punched speaker openings in the door panels are 115mm (4.5") in diameter, and the mounting holes are roughly 85mm apart... so with those dimensions you can start shopping around for something that fits.
When I did mine (years ago) the only simple, clean installation was with Alpine SPS-1329S speakers. They're actually pretty big but don't need a big or deep space. They suit <span style="font-style: italic">my</span> needs fine, but there were alternatives from JBL that had additional tweeters that needed to be mounted elsewhere - ideally up high nearer eye-level. I suspect there are similar options today.
There's always the option of putting them in boxes on the back shelf or hacking up the rear bulkhead - neither of which appeal to me.
One other thing that may or may not be an issue depending on your skill level with electronics: there's no real ready-made wiring harness available for the MG. Crutchfield, Best Buy, and others, usually rely on pre-made adapters that adapt the radios to the manufacturer's wiring harness. You pretty much have to make your own with the MG. I made mine with a 9-pin Molex connector so the speakers and power circuits terminate on the Molex, then there's an adapter that connects the radio to that molex inside the car. It'll make life a lot easier later if you ever need to update or change the radio.
I was also pretty leery of the wiring in the car. The original car wiring (if I recall correctly) had the radio power switching directly through the ignition switch - in other words, the ignition switch doesn't drive a relay, the radio's power actually flows through the switch. Not really knowing what the specs were, I reasoned that the current draw from a modern radio (and possibly amp) might exceed what they originally designed the circuit for.
I eliminated the ignition-switch completely and use a rocker switch that drives a 30-amp headlight relay, which in turn controls the radio. My plan was to run a separate power circuit from the battery to the relay (fused at the battery) with the radio and amp power coming from the relay. That isolates everything else in the car and doesn't stress the rocker switch. You could always drive the relay with the original ignition switch and the results would be the same (negligible power flow through the switch). The memory "keep alive" power is drawn from one of the plentiful always-hot wires (I think I split mine off the cigarette lighter).
I've never actually added the amp or the extra wiring to the battery, but with my wiring harness the mods would be pretty easy to do when I get to it...and as it is, I can switch the radio on and off without having the key in the ignition.