There are various alternatives here.Some level of buy-in from the players isn't that unreasonable a request. One assumes there's already buy-in to the setting, game system, and so forth otherwise those players (most likely) wouldn't be at that table, and it's not a big jump from there to hope for some buy-in to a story idea
One is to get the players to present the GM with a story idea. This can be done in various ways using various devices that may be better or worse fits for various systems.
In this thread I already mentioned a BW session where the player had chosen for his PC the belief I'm not leaving Hardby without gaining some magical item to use against my brother. So I started the session with the PC at a bazaar in Hardby where a peddler was offering to sell various curios and trinkets, including an angel feather from the bright desert.
A more player-driven technique to presenting the GM with a story idea is to use "kickers". Here's my take on that technique and some examples, from the first session of 4e Dark Sun:
The first half or more of the session was spent on PC building (despite my admonition to the players that they could only have 1 hour). With three players, we got 3 PCs: an eladrin bard with the virtue of cunning (with the Veiled Alliance theme); a mul battlemind gladiator (with the gladiator theme and wielding a battle axe); and a half-giant barbarian gladiator (with the wilder theme and wielding a glaive).
As the final part of PC building, and trying to channel a bit of indie spirit, I asked the players to come up with "kickers" for their PCs.
From The Forge, here is one person's definition of a kicker:
A Kicker is a term used in Sorcerer for the "event or realization that your character has experienced just before play begins."
For the player, the Kicker is what propels the character into the game, as well as the thing that hooks the player and makes him or her say, "Damn! I can't wait to play this character!"
It's also the thing that the player hopes to resolve at the end of the game. At the start of the next game with the same character, the resolution of the Kicker alters the character in some way, allowing the player to re-write the character to reflect changes.
In my case, I was mostly focused on the first of those things: an event or realisation that the character has experienced just before play begins, which thereby propels the character into the game. The main constraint I imposed was: your kicker somehow has to locate you within Tyr in the context of the Sorcerer-King having been overthrown. The reason for this constraint was (i) I want to be able to use the 4e campaign books, and (ii) D&D relies pretty heavily on group play, and so I didn't want the PCs to be too separated spatially or temporally.
The player of the barbarian came up with something first. Paraphrasing slightly, it went like this:
I was about to cut his head of in the arena, to the adulation of the crowd, when the announcement came that the Sorcerer-King was dead, and they all looked away.
So that answered the question that another player had asked, namely, how long since the Sorcerer-King's overthrow: it's just happened.
The other gladiator - whose name is "Twenty-nine", that being his number on the inventory of slaves owned by his master - had been mulling over (no pun intended) something about his master having been killed, and so we settled on the following:
I came back from the slave's privies, ready to receive my master's admonition to do a good job before I went out into the arena. But when I got back to the pen my master was dead. So I took the purse with 14 gp from his belt.
(The 14 gp was the character's change after spending his starting money on gear.)
Discussion of PC backgrounds and the like had already established that the eladrin was an envoy from The Lands Within The Wind, aiming to link up with the Veiled Alliance and thereby to take steps to save his homeland from the consequences of defiling. So his kicker was
My veiled alliance contact is killed in front of me as we are about to meet.
(A lot of death accompanying the revolution!)
With all that in place, we started the session proper.
As well as player-generated story ideas, it is also possible to have this be determined via random rolls. This is how our Classic Traveller game started, as per my post not far upthread: random generation of PCs, starting world, a few other worlds and a patron all suggest a starting situation, and that can then be built on during play. To reiterate a bit, in our CT game relevant aspects of the starting situation included a PC with a navy background and working as a medic, a marine officer patron, a high-tech world, a dilettante PC who had just won a starship in a bet, a PC spy, and a likely destination world with a disease-riddled atmosphere. It was this combination of elements - random generation and then riffing on that - that propelled the PCs into a collision with bioweapons conspirators.
In all these cases, of course, nothing will happen if the players aren't intereseted in playing their PCs. But there's no requirement for them to buy into a story idea that is presented to them by the GM.