If you break one spoke, it's generally worth replacing the one spoke, if you break a second, replace it as well, if you break a third, you need to replace them all.....

The first step is to examine the rim well, to see if it's cracked or bent. 1990 is awfully late for 27" a size that was common up until the late 1970's and fell out of fashion in the early 1980's being replaced by 700C... The real difference is that the rim on a 700C is 622mm in diameter, while 27" is 630mm in diameter. If you have a metric ruler, simply measure the slot on the brake from the bolt to the centre of the wheel end, if you have more then 4mm, your good.

Now examine the hubs, they need to be disassembled and the cups need to be examined, if they are pitted or scored outside of the running area, then the hubs will need to be replaced. You know the spokes will need replacing, so if the hubs or the rim also need replacing, just replace the wheel. If a 700C will fit, go with the 700C, there are more wheel and tire choices in that size. If your replacing the rear wheel, you may wish to replace the front as well.

You need to take the bike to the shop, because they need some information to match up axle type, axle width and gear count. Gear count is important, if it's 7 or fewer it's likely a freewheel, which is screwed on. If it's 8 speed or higher, it's probably a cassette, there were 8 speed freewheels, but they don't matter, because the gear spacing is the same on an 8 speed freewheel and cassette, but it will mean buying a cassette, instead of using your existing freewheel. If you do need to buy a cassette, you should replace the chain as well. Replacing the wheels is likely to leave your wallet about $200 lighter.