In my opinion, yes, you can hunt elk with the right 117 - 120 grain ammunition. That means premium bullets like the Nosler Partition and Accubond or the CT Technologies Partition Gold or Fail Safe. Sierra also makes a really good HP in 120 grain, and Hornady is supposed to be coming out with a .257 Interbond in the near future. If these are even available to you and you hand load or can afford the primo ammo... I also own a Ruger .25-06 and have had the same question. Happened to be in a gunshop when an old timer told me about his experiences using a .25-06 on Elk. Made me a believer.
There are better calibers for it. 30-06 to start, 7mm Rem Mag and
7mm WSM also. Don't forget the .280 Rem. Even though its possible, doesn't mean that it is a good idea. I like one shot kills that cause as little suffering as possible. Depending on your immediate hunting situation re: terrain, animal size, rut status, etc. , a .25-06 may be undergunned for your needs. However, it can be done.