The main ingredient in gunge is a food thickner. There are a number of options open to you.
If you are in the UK Natrosol HR250 (often mis-spelt Natrasol) from The Basic Chemical Company (see address below) is a good material. This is the supplier the BBC have used (not sure if they still do, but I wouldn't be surprised)- You will need to buy a minimum of 1 kg of the natrosol, but this makes a good gunge.
The Basic Chemical Company (UK) Ltd
Sands Industrial Estate
Ebay/ other websites
There are other suppliers that well sell you smaller quantities, either found on the web or through Ebay. Personally I don't believe these generally offer particularly good value, unless you only want to make one bucket full. It is likely (although I stand to be corrected) that these suppliers are selling on re-packaged Natrosol .
Xanthan Gum- this is sometimes available from the supermarkets (certainly Sainsburys) or health food shops as it is used as a gluten replacement.
Guar gum- another alternative that can sometimes be found in health food shops.
If you are in the US, you can use Methylcellulose (Methocel) from The Chemistry store- www.thechemistrystore.com or Ethylcellulose from Douglas and Sturgess- www.artstuf.com
Elsewhere you may need to find a local source for one of the thickeners.
The substances used to make gunge generally have long shelf lives (normally several years) and so can be bought in larger quantities and stored if desired. Moisture is their biggest enemy, so they will need to be packaged in a well sealed container.
You will also need
powder or poster paint
Suitable container to mix in- bucket for the quantity described
Wooden spoon, whisk, paint stirrer or similar
Step 1: Safety
The largest single issue is the slipperiness of the gunge- it can be very easy to slip over on. It is therefore essential to consider this when planning any activities involving gunge, including clean up and washing off as the most significant effects will be found on a smooth surface such as a bath!
Most of the materials are supplied as relatively fine powders, and as such create nuisance dust. It is therefore worth wearing a dust mask when handling large quantities. Likewise if this dust gets into eyes it can cause irritation so goggles may be appropriate.
Most of the materials are sold as safe for skin contact and to date I have not had any problems with this. I would however be slightly wary of allowing prolonged skin contact with the colourants as it may stain!
All of these main materials are not hazardous when eaten, however many are used as laxative additives, so it would probably be unwise to consume large quantities !
In addition to the specific information above, general safe chemical handling practice will further reduce the risk of any issues, in particular;
Store materials in sealed, well marked, containers in an area out of reach of children and pets
Wash hands before and after handling these materials
Dispose of excess/ used materials/ empty packaging responsibly
Wear dust mask when handling powders
Clean up any spills quickly- Avoid using water to stop the area becoming slippery
Do not use utensils/ containers/ cleaning cloths that will be later used for food
Supervise children if they are making the gunge