chainsaws are really quite a modern invention - it wasn't that long ago that all tree work was done by hand - we have run quite a few training days for people who want to use only traditional hand tools - and no modern kit - but after a day or two of really hard graft (especially after an hour or two on a double handled saw) many walk away and buy a chainsaw when they realise the time and effort needed to work with just hand tools.
so it is possible to use no machinery at all - but i think your biggest problem might be finding somebody who is prepared to work in this way - and i'm guessing you'd have to pay them to carry out the work, as the timber alone might not be incentive enough