Fabric

To do this you will need a square of cloth. You can buy one or make your own

Make your own
Decide what size.
50cm x 50cm Cloth for smaller items
A wine bottle will need 100cm x 100cm version.
Pick your fabric. It needs to be thin enough to tie but not see through obvs!
Hem the edges for a neater look. Or pretend frayed is chic.
For more detailed instructions try the lovely Omiyage Blog

Ready Made
For teeny tiny gifts use a hankerchief.
For bigger gifts, buy a headscarf. You can find loads in charity shops.
Smaller stuff you can use a bandana.

If you really cannot get any of the above locally I have listed some here!

Wrapping

Then use it to wrap your presents using this ancient Japanese technique of furoshiki. The following image is from the Evermine Blog which is full of lots more great ideas.

If you do go down the paper route, be sure to use this plastic free sticking tape