I would actually restrict things further, since introducing infinity can lead to unintuitive results orthogonal to computability ;)

Rather than 'possessing' an infinite tape and infinite time, I prefer to think of turing machines as always being allowed to perform one more step, and always being allowed to use one more tape cell. I find this characterisation of time quite intuitive as-is, whilst the tape can be made intuitive by imagining tape factories on each end, which produce more cells whenever the read/write head gets close.

In fact, such "tape factories" have been used to build universal turing machines in the game of life http://rendell-attic.org/gol/fullutm/index.htm