We used to call this a Ginger Beer Plant, I've copied instructions from a BBC site below, but I'm fairly sure we didnt store it before drinking it! I don't think we used the lemon either, and would have used ordinary granulated sugar instead of caster because it's cheaper.
Ingredients:
25g Dry yeast
2tbsp ground ginger
2tbsp caster sugar
½pt (300ml) water
Method:
Stir the yeast in with ¼ pint (150ml) of lukewarm water and a bit of sugar. Leave this mixture for 10-15 mins.
Mix the wettened yeast with the sugar until they cream to form a liquid. Then add the ground ginger and remaining water. Stir this mixture well and pour it into a jar with a loose-fitting lid, so as to let the mixture breathe. Now leave this mixture in a cool place.
The next day, and each day thereafter, add 1 level tsp (5ml) of ground ginger and the same amount of caster sugar to your jar and stir the mixture thoroughly.
After 10 days. dissolve 18oz (500g) of caster sugar and 1½pt (900ml) of water, bring this mix to the boil and let it cool slightly. Add the strained juice of two lemons to the water.
Strain your ginger plant through fine muslin and add the strained liquid to the sugar and lemon juice mixture, along with 6pt (3.4l)water.
Stir the mixture well and bottle it straight away in strong screw-topped bottles, like those in which you would store cider or beer. Make sure you store the bottles in a cool place for about two weeks before you drink it. This is essential!
To Make More Ginger Beer
Halve the sediment left on the muslin and divide it into two separate jars. Add ½pt (300ml) water, 2tsps (10ml) and 2tsps (10ml) caster sugar to each jar. Stir it well and continue the same process as before for ten days.
This means that the produce is doubled every ten days.
Here's a similar one in pdf format:
http://www.microbiologyonline.org.uk/forms/gbeer.pdf