Hungary uses Forint (HUF) as national currency however Euros are widely accepted but with high charges or exchange rates.
In most places you will find bank cash machines that dispenses only in Hungarian forints, but there are OTP bank and Euronet cash machines that dispenses also EUR currency in Budapest.
OTP Bank (euro)
Deak Ferenc utca 7-9, Budapest
OTP bank (euro)
Baber utca 9
https://www.otpbank.hu/portal/en/BranchATMSearch (please select detailed search and select EUR)
There are number of EURONET ATM cash machines through out Budapest for drawing out Euros. You can locate ATM’s here: https://www.euronet.hu/en/search