Working in Oman requires that you hold a residence permit. In common with other Gulf countries, you must be sponsored by an employer to obtain a residence permit. It's not uncommon for people to enter on a tourist visa then look for a job - this is fine. Penalties for the employer are substantial if they are caught employing illegals, although this naturally varies depending on how good their connections are.
The majority of positions are filled by expats from the sub-continent. Positions for Europeans tend to be restricted to upper management levels or specialised occupations, so don't expect to pick up a position as you pass through unless you are prepared to work for very little!