As other have indicated, the gym has a cast iron case for getting their money through court action.
The court will issue a County Court Judgement against you. The register of CCJs is a public document; anyone can check it to see if you've had a judgement made against you. (It's available online here:
http://www.trustonline.org.uk/ ). Credit reference agencies refer to that register when checking your creditworthiness. If you get a CCJ you'll almost certainly have all credit applications refused until some time after the judgement is 'satisfied' (i.e. you've paid all of the money owed).
However the courts are very understanding when people are having genuine difficulties making payments. As long as you explain your situation to the court you'll be ordered to pay the money in small instalments (possibly over several years), rather than all at once.
As long as you comply with the court order you'll have nothing else to worry about. If you fail to comply with the order the gym can seek enforcement. The usual ways of doing this are either to get a further court order forcing your employer to pay money directly to them, or to get an order allowing them to send bailiffs into your home to seize your goods.
Chris