Is there any way to change victory conditions with the game already started?

I made a simple web app where you can change victory conditions and some other save parameters: Civilization V save editor

I created that app because Oak's answer has two problems:

1. It disables Steam achievements for this game, if that's something you care about
2. It won't work on Mac or Linux since the Civ V SDK is only available for Windows
   
   

Here's my original answer where I detailed how to edit the save file by hand (not for the faint of heart):

1. Open the save file in a hex editor of your choice
2. Search for this text, which should be somewhere around address 26a0: TXT_KEY_TURNTIMER
3. About 60 bytes later you should see a byte containing 40. Just before this are the 5 bytes (10 hex characters) that set the victory conditions: Each one will either be set to 00 for disabled or 01 for enabled, and as best as I can tell they're in the same order as they are in the UI:
   1. Time
   2. Science
   3. Domination
   4. Cultural
   5. Diplomatic
4. Edit the values to enable/disable the victory conditions as desired. For example, if you want to enable only domination and diplomatic victories, the values just before 40 would be: 00 00 01 00 01
5. If you modify time victory, you will also need to modify max turns:
   1. Search the save file again, this time for the text \Maps. This text will occur multiple times; the final occurrence is the one we want. It should be somewhere around address 1ad0.
   2. At the end of the text, there will be 4 bytes with all zeros (00 00 00 00). The 2 bytes (4 hex characters) immediately following set the max turns: The values are stored little-endian, so you have to reverse them to convert from hex to decimal.
   3. Set max turns as appropriate:
      • If you wish to disable time victory, set max turns to 00 00
      • If you wish to enable time victory, set max turns to the desired value. Here are the default values for the various game speeds:
        • 4a 01 (0x14a): 330 turns (quick)
        • f4 01 (0x1f4): 500 turns (standard)
        • ee 02 (0x2ee): 750 turns (epic)
        • dc 05 (0x5dc): 1500 turns (marathon)
6. Save your changes and open your save within the game
7. To make sure it worked, at the top right under Turn click the scroll icon > Victory Progress. It should reflect the changes you've made:

There is no way to do that with the base game.

You can do that by using a custom FireTuner panel, as detailed in this thread on civfanatics.com. There's a download link for the panel there, as well as information on how to add this panel to FireTuner and how to use it.

There's also some useful information on that thread for those new to FireTuner:

how i can open/start the tuner????

1. Download "Sid Meier's Civilization V SDK" from Steam (through Library -> Tools)
2. Open up the config.ini file located in your Civ5 user folder (e.g. "C:\Users\Username\Documents\My Games\Sid Meier's Civilization V" on Vista)
3. Look for the line "EnableTuner = 0" and change it to be "EnableTuner = 1" then save changes.
4. Launch Civ V
5. Launch the Tuner from the SDK shortcut of the Steam menu

Be aware that changing victory conditions mid-game might seriously screw with the AI.