No; not all contracts need to be in writing.
In some cases, statute law requires contracts to be in writing (e.g. contracts to buy and sell land, to buy a car, and door-to-door sales contracts). However, it is always useful to write down the terms agreed to by the parties so that there is less potential for dispute about what was agreed.
A written contract can be drawn up by writing down all the terms agreed to by the parties and getting each party to sign and date the document, either in person or electronically. You do not need to use legal or formal language in a contract for it to be enforceable. The most important thing is that the language used in the contract is clear and precise so that both parties understand what is being agreed.
It is useful to keep any documents relating to the agreement (e.g. quotations, invoices, emails and receipts), so that if a dispute does arise, these documents can assist to resolve any differences between the parties.