Oman has several high class medical facilities on par with western nations and tend to be affordable even if you don't have insurance.
However, should you do need medical services and lack insurance, you may be prevented from leaving the country until your debt is paid.
Travellers in Oman are at risk of insect borne diseases such as Malaria, dengue fever and leishmaniasis, but most of these are relatively under control. But you should be especially attentive on the Musandam Peninsula. Take general precautions such as the long clothing everyone else seems to wear (probably a reason for that) and make sure you have some insect repellent for exposed areas. For tips on bite prevention, read this.
Of course, the most common health problem travellers face is simple the extreme heat. For people not used to such temperatures (54 Celcius is common), especially in the summer months, it is very easy to become dehydrated, sunburnt, or sunstruck.
Be sensible, stay out of the sun as much as possible, use sunscreen, and make sure you drink plenty of water. A warning sign of dehydration and sunstroke is a headache that increases in severity starting at the temples and behind the eye sockets. If you experience that, start drinking more water and take some hydrating salts if needed.
The water quality in Oman is actually quite good most of the time, but even the locals drink bottled water and you should probably do the same. Likewise, avoid ice cubes as they tend to come straight from tap water and as in most places, avoid raw or undercooked food and unpasturised milk products.
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