bet365, Paddy Power, Unibet and Betfair are rated as the best betting sites available to Irish residents by the OLBG ratings. Of these Paddy Power is the well known Irish based company. Boylesports and Quinnbet are other Irish based companies with online betting sites available but aren't so highly rated by OLBG users.
It is perfectly legal for Irish punters to place sports bets at a licensed and regulated offshore online sportsbook. In fact, the Republic of Ireland welcomes international online sportsbooks into the fold through their licensing policies and taxation schemes.