How can a website accept payments from users in South Asian countries like Bangladesh or Pakistan?

Your only option is to tie up locally (via opening a Nostro account) with any of the banks here.

By opening of a nostro account, you can ask your clients for a direct deposit of funds to your nostro account. A terminology known here as IBFT (Inter Bank Funds Transfer) for those who are already holding a bank account and want to pay directly to you (without using their credit card - if they have one).

Nostro accounts can be opened without you being physically present here. However, it does require a substantial amount of paperwork. If you are a small time merchant, I wouldn't even try to apply. Nostro accounts have to be approved and validated by the central bank of the country concerned.

A twist to the same situation is to partner up (alliance) with a local merchant and use their payment mechanism to divert funds to you (i.e. use their bank account for deposit of funds, or cash amount pickup). Needless to say, you will have to split some of the profits or pay a charge for this service.

The latter approach would be much easier.