Since I have not read the manga yet, I am just going by what I saw on the anime for their information.  I am sure that the manga goes much more in depth about their pasts and how they met and everything, but I can only go by what I've seen so far.

Ban Midou grew up with an extremely powerful ability known as the Jagon.  With it, he could cause a person to enter a dream like state that would last for one minute just by looking in their eyes.  Ban used to work with a girl named Himiko and her brother and counted them among his closest of friends.  However, tragedy struck when he was forced to kill her brother when a disease that he had reached its peak.  He literally begged Ban to end his life and so Ban had no other choice.  It was hinted that Himiko might eventually suffer the same fate and Ban took it on himself to try and keep a close eye on her, even though she never forgave him for killing her brother.

Ginji Amano grew up an orphan in the Limitless Fortress, a place that was virtually a society within itself.  Battles and magic were common and Ginji lost a lot of friends during their struggle to survive.  As he got older he discovered his incredible magical powers which he could use to harness the power of electricity.  He became a powerful leader in the Limitless Fortress known by all as the Lightening Emperor.  He formed a group called the Volts of which his two friends Shido and Kazuki were members.  Everyone looked up to Ginji and relied on him and his powers to protect them all.

As time went on, Ginji became more and more afraid of himself and his Lightening Emperor persona.  He felt that he was losing his true self every time he transformed into him and felt that he was putting everyone in danger more than helping them.  He chose to leave the Limitless Fortress before he caused the deaths of any more of his friends.  He met Ban when the two apparently fought against each other (it wasn't that clear in the anime what truly happened), and Ban helped Ginji come to terms with leaving.  The two left together and Ginji suggested that they work together to use their powers to help others.  And thus the Get Backers recovery service was born.

The Get Backers promised to get back anything that had been taken from someone.  Ban and Ginji worked well together in their business, although they never had much luck with money.  They had a near 100% success rate and were able to use their powers for the good of others.  They soon became inseparable and relied on each other to get through everything.  Ban was extremely protective of Ginji and put all his trust in him, no matter what.  And whenever Ginji found himself in trouble, he always let his thoughts drift to Ban, for he knew that he would always be there for him.  Ban and Ginji even seemed to have a spiritual connection of sorts, for Ban could feel when Ginji was in danger.  They had a very cute friendship and a possible romance was hinted at many times.  So even though they were not an "official" couple, there was more than enough subtext to keep the Ban/Ginji fans happy.

In the final episode of "Get Backers", Ban and Ginji were off on another recovery service job, so it is probably safe to say that they would continue that together.  There really is no reason that I can see that would force them apart, as they already told each other that they would never leave the other one alone.  And that lingering gaze they shared in the car before Ban pulled Ginji close leads me to believe that their relationship is not far off, if it hadn't already been going on at that time.

This scene was from when Ban was trying to share his past with Ginji as they were pushing their car down the road.  Ban was having trouble putting his thoughts into words and Ginji told him that it was okay if he didn't tell him.

Ban: "See... I, uh... Her big brother and I, we, uh..."

Ginji: "It's okay, Ban. And you know why it's okay? 'Cause I believe in you. Wasn't it you who told me 'you know what the S in Get Backers means?' It means you're never going to have to be on your own. [Ban smiles] So always remember, you don't have to carry the weight of the world on your shoulders there- ah!" [Ban starts pushing the car Ginji was leaning on and Ginji falls over backwards]

Ban: "You just sleep there buddy, I'll get the platinum on my own!"

Ginji [gets up and runs after Ban]: "Hey! Hey, Ban! That... That's mean! Remember the S in Get Backers means-!"

Ban: "Shut up! Money is a whole different issue!"

Ginji: "That is so not fair!"

~~~

This quote was from the episode "Gathering, Retrieval Team" when Ban and Ginji had finally met up again after being separated for several episodes.

[Ginji sees Ban and gasps]

Ban: "Yo, Ginji! You look like you're doing okay, huh?"

Ginji [rushes over and lifts and spins Ban in the air]: "Ban! Ban! Ban! Ban! Ban! [he giggles and Ban ruffles Ginji's hair before Ginji hugs him]  "I'm so happy, so happy, so happy to see you alive and okay!"

Ban [smiling]: "Ha! For crying out loud, stop being so dramatic."

Ginji: "But, but..."

~~~

After just being reunited, this quote from the same episode shows Ginji's reluctance to part with his best friend again.  The group all rolled a die to determine which room each person would go into.

Ginji [looking at the die he rolled]: "Five, huh? That means Ban has to roll next, doesn't it? That doesn't mean Ban and I have to split up again, does it? That's not fair! We finally found each other and now we have to say good-bye! What should we do, Ban?"

Ban: "We shut up and roll."

~~~

From the same episode as the two above, Ban stops Ginji before they go their separate ways again.  He makes sure that Ginji knows that he will be there if he needs him.

Ban: "Ginji."

Ginji: "Huh?"

Ban: "If things get bad, you just call me. You got it, pal?"

Ginji: "I'm telling you, it's okay. You can be such a worrywart sometimes, Ban, I swear!"

Ban [sternly]: "Ginji."

Ginji: "It'll be okay, Ban. You know why? Because we're the Get Backers, that's why."

[they smile at each other]

~~~

This quote is from the episode "Sword of Revenge - Ban vs. Fudou".  Ban and Ginji had been apart for awhile again after going into different paths in the Limitless Fortress. Ban flashed back to a memory in his past and then freaks out when he senses Ginji is in danger.

Old lady: "Remember this part. Nothing in this world is unconnected to the movement of the stars in the sky. Therefore, invoke the constellation of your fate when you have something you truly desire. Something you don't want to lose."

Ban: "Damn. I went and remembered something I didn't want to."

[Fudou comes at him to attack and a bolt of lightning appears]

Ban: "The lightening must have shorted out the VR!  [he sees Ginji and speaks the next line in his head] Ginji! You've changed, haven't you?"

[Fudou says he can see Ban's death and Ban begins chanting]

Ban [in his mind while invoking the serpent]: "Ginji, you're not like that! You're not me! Ginji!!"

[Ban finds the strength to attack Fudou]

Ban: "Out of my way!"

~~~

In the episode "Final Battle! Ginji vs. Makubex", Kazuki notices how much trust Ban has placed in Ginji.

Kazuki: "Ban here believes in Mr. Ginji with all his big heart."

[Ban smiles]

~~~

In the episode "Get Back the Eternal Goddess", Ban explains to Akabane that he has a connection with Ginji.

Ban: "I know in my heart when something bad happens to him."