I agree with most of the posters about duration.
That is, for longer runs, where the definition of longer is very personal, water at least for hydration is required. Water bottle holders come in all shapes and sizes now.
- Fanny packs, which can irritate the small of your back where the bottle top bounces into it.
- Angled bottle fanny packs, which try to alleviate that pain.
- Fuel Belt and copycats where there are many small bottles on a belt.
Everyone is different, and hydration requirements will differ.
You can get an estimate of how much you sweat, and thus need to replace, by starting a workout, fresh, weigh yourself, do your workout, no drinking, for a reasonable amount of time (an hour works well) and then shower, and weigh yourself clean after.
Since you cannot appreciably burn off significant mass in an hour of exersize, you can estimate how much you sweat out. A litre weighs about a kilo, (which is why Metric is so useful).
Worst case is usually 1 to 1.5 litres an hour for heavy sweaters. If all you are doing is an hour or three then you cannot dig youself too deep into a hole. However for longer stuff like marathon and long course triathlons, it is important to stay ahead of the drinking and not get into the hole, since you will not be able to get out of it during the race.
I am at the heavier end of that sweating scale, and on my first Ironman drank 18 litres on the bike, 8 litres on the run, and lost 7 lbs. It was a bit extreme.