What country and/or state's phone numbers begin with the digits "08"?

Ireland's country code is 08.

I don't see any country code of 08.

http://www.1areacodescountrycodes.com/all-country-code-numbers-countrycodes.htm

No country has an International code starting with 0.

0 is widely used as the access code for non-local numbers - i.e. in Southport we dial a local number such as 456789 as 456789 where anyone outside the Southport group of exchanges (01704) would dial 01704456789. Dial an 0 and you get to the part of the machine dealing with non-local, dial another 0 and you're into the International section and add on the International code for the country, say, 382 for Montenegro. International codes are often written as +382, which means use whatever the country you're in uses to access International and then 382.

08 is the Australian internal code for South Australia, Western Australia, and the Northern Territory.
http://www.idd.com.au/

When dialling in from abroad you will probably have to drop the 0, as you do in the UK - my Southport example would be +44704456789 (no, it's not my number - it doesn't even exist at this date).