I know that sailors often drank diluted rum for hydration because freshwater was difficult to keep on a ship. How strong was the rum? Could the sailors easily get drunk from drinking it?

Ah, the question of "what was rum in the old days like"? I've heard it asked a few times. Let's go to Feeding Nelson's Navy: The True Story of Food at Sea in the Georgian Era by Janet Macdonald on pg 42:

"The strengths of the brandy and arrack [back in the 18th and early 19th century] are not known, nor with any accuracy is that of the rum of the day, but it is believed to have been at least four times as strong as the 'Navy' rum sold today; that official half-pint would have been the equivalent of two modern bottles.

For context, a "modern bottle" of "Navy rum" contains 40% alcohol, give or take a couple percentage points. It should be explained that during this era, the British sailor's beer ration was a gallon a day. This beer was a 2-3% alcohol content, similar to the stuff ashore called "small" beer. But since beer doesn't keep well in warm climates and even then could go sour in the not so clean casks in a matter of days, substitutes were made at various places when beer ran out or couldn't be provided (though beer is prioritized since taxes on it were lower in England). In northern waters in Europe, a pint of wine often fortified in brandy became the substitute for the gallon of beer. In the Mediterranean, it was a half pint of brandy. In the East Indies it was a half pint of arrack. Finally, a half pint of rum was the norm for the West Indies.

But I can hear the cries of "oh, what about grog?" Grog is not just rum-based (though is the most well known mix of water, a spirit, and eventually other things like lime). The book I am using for a source further explains (on the same page as above):

The requirement to dilute spirits is thought to have originated with Admiral Edward Vernorn, in the West Indies in 1740. He ordered the rum ration to be diluted with a quart of water [per half pint], this to be done on the open deck in a scuttle butt kept for that purpose, in the presence of the lieutenant of the watch, and of course any of the crew who cared to observe the process to make sure they were not given short-measure. Admiral Vernon was known as 'Old Grog' from his habit of wearing a cloak made of grosgrain (a heavy corded silk fabric) and his rum and water mixture came to be known by the same name. Later, on the advice of Doctor Trotter, lemon or lime juice was added to the mixture as an antiscorbutic. Rum has come to be known as 'Nelson's blood', perhaps by people under the impression that his body was preserved in rum; it was actually brought home in a cask of brandy, as evidenced by a letter from the Victory's purser Walter Burke, asking to be allowed credit for this in his accounts.

Since the sailor had to consume a lot of calories to do the work required on a ship (taking in between 4,000 to 5,000 calories a day, pretty much what a modern American football player might consume for training periods) and preserve against the potential cold that could result from weather or just the cold caused by wearing wet clothing - besides the alcohol providing some calories, it also helped make safe the liquids needed for hydration (and one can easily imagine how much sweating in such work would also work up a need for re-hydration, and alcohol can do that).