Start Freelancer as Pirate
Speaking strictly about single player vanilla Freelancer you can't. You have to play the story at least as long until you have access to the jumpgates. If that is too much story for then that means: NO.
How to get around those restrictions
You can however play multiplayer (even if it's just a server for you) or install appropriate mods to lift those initial restrictions.
As for your apparent problem with faction standing: Every faction has friends and enemies. By shooting members of one faction you not only anger them and their friends but also please their enemies. Determine which enemies your favorite faction has and shoot them (For pirates try shooting the police). It takes a bit of grinding and a good enough combat ship but it is feasible.
Further suggestions
If you are interested in mp may I suggest you check out the Mod "Shattered Worlds - War Torn".
You can use the open single player mod.
In vanilla, you can become a pirate even without jump gate access and - if you're new to the game - it can be rather fun. In vanilla, Rochester (the Junker base in the Jersey debris field) sells the Bloodhound Pirate Light Fighter (that used by the Rogues) while Buffalo (the Rogue base in the badlands) sells the Dagger Border Worlds Light Fighter (that used by the Outcasts). You can acquire either ship, if you like, and pirate the trade lanes at will. The only downfall is that you'll make yourself hostile to most house police forces, navies, and corporations, but this is not a bar to playing the storyline.
You are able to use docking rings whether or not friendly with the faction that controls them, and the campaign tends to unlock those bases arbitrarily that you need to dock at for the purpose of continuing the storyline (even if these are unavailable while in freeplay). I played through the first New York segment of the game (up until you and Juni leave for Bretonnia) as a pirate many times myself, completely hostile with the liberty police, navy, and corporations by the time I left (which didn't really matter because] they turn hostile to you with the storyline anyway).
As per Rochester not having missions, both Rochester and Buffalo have missions IF your rep' with the junkers and rogues, respectively, is high enough. One good way to advance your rep' with the junkers and rogues, while not damaging your rep with house forces and corporations, is by killing Xenos - pirates themselves but also enemies to virtually all other pirate factions.