Diagnosis
Total power loss. This means it's not getting the 24VAC from your HVAC system's transformer. The issue is either at the furnace or a blown fuse on the furnace control board.
How to Fix It
Go to your furnace and shut off power (at the unit and the breaker). Open the panel and locate the control board. Look for a small, clear 3-5 amp automotive-style fuse. Pull it out and check if the filament inside is broken. If it's blown, replace it with an identical amperage fuse. Restore power and see if the ecobee boots.
If That Didn't Work
With the power OFF, double-check all your thermostat wire connections at the furnace control board. A loose wire on the 'R' or 'C' terminal can cause this. Ensure each is snug under its screw and not touching another terminal.
Heads Up
If the new fuse blows immediately when you restore power, you have a short in the low-voltage wiring (usually a pinched wire where it enters the furnace). That's a bigger job and might need an HVAC tech.