The Toffees went closest to breaking the deadlock in the first period when Timothy Castagne was forced to make a goalline clearance after James Tarkowski's shot had hit the bar.
The woodwork was struck at both ends of the pitch in the second period, as the two sides failed in their quest to find a winner.
As a result of the draw, Fulham are now winless in five competitive matches, while the visitors have now gone four away matches without a victory.
The first chance of the contest fell the way of Everton's Jack Harrison, who could only fire wide after being neatly teed up by Dominic Calvert-Lewin.
Fulham then offered their own early threat through Antonee Robinson, with the full-back forcing Jordan Pickford into a routine save with a deflected right-foot strike.
As the first half reached the halfway point, Everton went agonisingly close to taking the lead after an Ashley Young free-kick caused havoc in the Fulham box.
Bernd Leno parried Young's driven set piece into the path of Tarkowski, who saw his effort strike flicked off Issa Diop's head and onto the crossbar, before the spinning ball was hooked away by Castagne.
Everton continued to search for the crucial breakthrough, as Arnaut Danjuma spurned a promising opportunity when he scooped the ball harmlessly over the bar
Fulham's Andreas Pereira then drilled an effort over Pickford's goal just moments before Raul Jimenez failed to hit the target from close range.
After going into the break on level terms, Calvert-Lewin was inches away from breaking the deadline within 10 minutes of the restart when he rose from a corner to head the ball against the frame of the goal.
At the opposite end of the pitch, it was Castagne who was left frustrated after his headed effort bounced off the crossbar.
As the contest entered the final 15 minutes, Pickford showcased his quick reflexes to pull off a superb save to thwart Tosin Adarabioyo.
Everton substitute Beto then nodded over from close range in stoppage time, ensuring the contest finished in a goalless draw.
While the results moves Fulham up to 12th position in the Premier League table, the Toffees drop into the relegation zone following Luton's 4-0 win over Brighton & Hove Albion.