Our 5 month old baby has has all runny stools 3 and has thrown up 3 times in the last 24 hours. he does not have a a fever. should we take him to the hospital?

Common issue: Is the kid taking in enough fluid value to wet diapers every 6-8 hrs ? Is the kid alert & willing to try feeding ? Does he smile some & want to play ? If so you can monitor this at home. If he doesn't wet in 8-12 hrs, won't feed or lays around looking bad a visit would be reasonable. The location would depend on what is available. In the US this would be seen in an urgent care clinic.

If he is lethargic or refuses to even try to feed, then he needs evaluation. Otherwise, I would try to get through this at home. Get some rehydration fluid such as Pedialyte, and start with just a teaspoon full every 5 minutes. If he keeps that down for 1-2 hours, slowly increase the amount. If you're nursing continue nursing small amounts at frequent time intervals.Monitor his status frequently