Do I have to have all four molex connectors connected to my Gigabyte GTX 970 in order to make it work

Solution
Agreed. Sounds like your power supply is under-wattage and old, and that's a bad combination. You're jeopardizing your new card. I recommend that you purchase a minimum of 650 watts, so that you have option of upgrading to a single 980 Ti in the future. If you might actually upgrade to two 980 Ti's in SLI in the future, then consider a 850 watt PSU that is tier 1 or 2. Check the resources below. Personally, I like EVGA GS and G2 power supplies, but the choice is ultimately yours.

http://www.realhardtechx.com/index_archivos/Page362.htm

http://www.tomshardware.co.uk/forum/id-2547993/psu-tier-list.html
ARe you using 4 molex to 6 or 8 pins connectors for your video card? If so, I would say get a new PSU. a 970 is an upper end card and requires good power. Using molexs means your psu doesn't have 6 or 8 pins and would probably means its a cheap psu.

what psu do you have.
 
Agreed. Sounds like your power supply is under-wattage and old, and that's a bad combination. You're jeopardizing your new card. I recommend that you purchase a minimum of 650 watts, so that you have option of upgrading to a single 980 Ti in the future. If you might actually upgrade to two 980 Ti's in SLI in the future, then consider a 850 watt PSU that is tier 1 or 2. Check the resources below. Personally, I like EVGA GS and G2 power supplies, but the choice is ultimately yours.

http://www.realhardtechx.com/index_archivos/Page362.htm

http://www.tomshardware.co.uk/forum/id-2547993/psu-tier-list.html
 
Solution
Agreed with others, if you need all 4 molex adapters for your PSU because it does not have any 6 or 8 pin pcie connectors then you need a new PSU.
Any PSU that does not have PCIE connectors was not made to support a graphics card like the 970.

FYI, just because they slapped a sticker on it that said 600w in absolutely no way means it was tested or required to output 600w (more likely it will fry itself trying to output 400)