If you become tempted to purchase a bach over the Christmas period (as a certain partner of ours did one year) then an agreement on an envelope will certainly be legally enforceable if done properly. Debra bought her family bach by recording an agreement on an envelope that included adoption of the standard ADLS/REINZ terms for the sale and purchase of real estate.
Where land is concerned, the agreement does need to be in writing to be enforceable. The Property Law Act 2007 requires that a full written record of the terms of the contract is made, and the contract is signed. To enforce a contract for the sale of land, one would have to generally produce a signed, written agreement or a written note of the terms between the parties, subject to limited exceptions. Prior to the Property Law Act 2007 a wider variety of dealings could create a sufficient memorandum of an agreement, e.g. a letter from a purchaser to a third party which contained an admission of the existence of a contract and description of its essential terms was enforceable against the purchaser. Now the wording does not seem to allow for anything less than a full record of all the terms of the contract, although that may be in more than one document.
The writing and signatures can all be in electronic form, provided the Electronic Transactions Act 2002 is complied with. That is contemplated by the standard form of agreement for sale and purchase which permits the sending of counterpart copies by e-mail to create the contract.
So, if you are tempted to buy a bach, make sure your agreement is conditional on a full due diligence being completed when you return as that agreement on the back of an envelope may well be enforceable.