Switching from a dérailleur to a gear hub I'm upgrading an old bike with a five speed dérailleur system which I'm trying to convert to use a three speed gearhub from Shimano. I have assembled the wheel, I know how to set up the gearhub itself, but there's one question that bothers me: the chain.

Currently, the dérailleur pulls the chain and keeps it in tension. When I install the gearhub wheel, the dérailleur itself is ought to go away, but as I understand I need to keep the chain tight. Should I make it shorter or install another pulley? Is it possible to somehow reconfigure the dérailleur to pull the chain as a temporary solution until I do it properly?
"Old" bikes – those from before the '80s or so – usually have horizontal dropouts that allow maybe 2 or 3 cm of fore and aft movement. That should give you enough room to tension the chain. I would think that a 5-speed bike would date from this era and be built with horizontal dropouts.

If that's the case then you can probably don't need a chain tensioner, just adjust the chain length so that it is long enough that you can get the wheel into the dropouts and short enough that you can get the tension right before you hit the back of the dropout.

Failing that, you could set up the existing dérailleur as a chain tensioner – either by leaving the shifter in place (and remembering to leave it alone) or by removing the shifter and cable and setting the limit screws to hold the  dérailleur in place. You can also buy chain tensioners that are just "unshiftable" dérailleurs.