The fifth badge is the Relic Badge won by defeating Fantina in the Hearthome City Gym. This gym is not available until the Old Charm is delivered to Celestic Town.
the fifth badge is from Sabrina in Saffron City.
The fith badge is the Hearthhome city badge.
Get the next badge
In Pokemon Red, the fifth badge is possessed by the Poison-Type Koga. The badge is named Soul Badge. It lets you use Surf outside of Battle.
Heathome City
fuchsia city
You need to defeat Koga in Fuchsia City to get the 5th badge, the Soul Badge.
Hearthome
to petalburg city
hearthome city
Fuschia city.
Petalburg City.