What exactly does "one-bar fire with 50p in the meter" mean?

Were there actually coin-operated electric heaters in the 80s in Britain?

It's actually very similar to that. It's not the electric heaters which were coin operated, but the whole electricity system for the house. You'd deposit money directly into your electricity meter which would then allow you to use power equal to that cost, and someone would come and collect the coins out of it periodically, it's called an Electricity prepayment meter, and it looks like this, though this is a slightly later model that only takes £1 coins, older versions would accept smaller denominations:

These were very popular for rented accommodation as it meant that if the tenant didn't pay their electricity bill, the property owner wouldn't be held responsible, as well as removing the need for tedious paperwork whenever the property changes hands.

They actually still exist in a very similar form even today, and my house had one until a few years ago, where instead of being coin operated, they had a form of electronic card which you would remove from the meter, take down to the local shops and "top up" with real money (the shops have a special machine for this), then you plug the card back into the meter and it adds the balance stored on the card onto your power allotment. These newer models look something like this:

So for exactly what that meant, the question is basically a shorthand of saying:

If you put 50p into your electricity prepayment meter, how long could you run an electric heater with a single bar (heating element) before your power shut off?

@Crow's answer is great. The only thing I'd add for clarification is that a one-bar heater is an electric heater using a single wire filament, typically wound around a single horizontal supporting bar (often something like a glass rod). You could have multiple elements stacked vertically, hence one-bar, two-bar etc). They've largely been superseded by other, more efficient, forms of space heating, e.g. halogen heaters.

The heating element would sit in front of a curved metal shroud to reflect radiant heat forward, and by the 80s would have had a protective metal grille in front of it. (In earlier decades, it might have been completely exposed!)

One bar would have typically used around 800W - 1kW of power. While imprecise, even Sir Humphrey would have had a sense of how much heat output you'd get from a single bar.

(Photo from https://www.flickr.com/photos/jocelynb/426268348, CC BY-NC 2.0 licence)

Along with electricity meters there were also individual bar heaters activated by a coin and which would have been installed in bedsits or rooms in university halls of residence where the user would not have control over the overall electricity supply.

As per the previous answer, the 50p coin would keep the heater running for a fixed amount of time. The fact that Sir Humphrey doesn't know how long this would be is being used as a sign that he is out of touch with the people.