70mph uses about 12kW of power on a motorcycle, to go 350miles at 70mph takes 5hours therefor 5 hours at 12kW is 60kWh
current battery technology weights about 150Wh / kg, so a 60kWh battery would weigh 400kg (880lbs).
Current (cheapest) battery technology costs about $0.5 / kWh, so a 60kWh battery would cost around $30,000.
Until battery (or other electrical energy storage) technology improves a huge amount you wont be seeing a motorcycle that con do 350miles at 70mph.
As far as I know there are no hybrid electric motorcycles out there for purchase, but if you wanted to make one, you would need it to have at least a 12kW engine to sustain 70mph. Something like a Ninja 250 engine would be able to deliver that. Building a an electric motorbike with about 5kWh of battery, a 50kWmotor, and a ninja 250 or cbr 250 engine that could also power the rear wheel would be your best bet and achieving your target.
-ryan