First Idea-

The players are immortals (various kinds) but also heroes. So for instance someone could be [URL=https://s-media-cache-ak0.pinimg.com/564x/39/ba/d6/39bad613af2e86471b72c35a0c2ab939.jpg]Batman[/URL] through the ages, while another could have been Daredevil or Green Arrow - who obviously started off as Robin Hood. So want to import your favourite hero, go right ahead. You all start with enough money to fund to have spares for your equipment. No billionaires - now where would be the fun in that.

Immortal types are-
Bio-Energy: They possess increased amounts of cellular energy rendering them not longer susceptible to cell decay, staying at the prime of life. You can regenerate lost limbs over a period of several days.
Cognitive:You can control your body by mind alone, you are immune to viruses, disease and most toxins. Due to this you are more intellectually capable than most others.
Genetic: You were born with recessive genes rendering you unaging, immune to all disease and viruses, but unable to regrow lost limbs.
Virus: Some kind of virus rendered you unable to age, but also you are immune to all toxins, disease, viruses, including nerve agents. You heal  about ten faster than a normal human.

All are approximately five times physically superior to humans.

Second Idea (Couresty of G3njii)-

The players are the next generation in heroes, taking over from the old - learning the history of those that came before while fighting against the crime in the world. 

All are able to select their proclivities in training:-

* Martial Art Master - You're Pei Mei or near as damn it
* Assassin - You're Jason Bourne or Natasha Romanov
* SpecOps - You're Frank Castle (MCU) or one of the Expendables
* Spy - You're Sidney Bristow
* Super Sleuth - You're Sherlock Holmes (Sherlock tv series)

You are tasked by a time traveller (recruited separately) to help protect the world. Power level is Arrow, and Legends of Tomorrow. So all heroes would start above normal, before the immortal template was added.