No, this isn't what He meant—although it's not necessarily wrong to refer to some problem we are having as a "cross" we must bear. In Jesus' day, a cross was a symbol of suffering, and we all have trials and afflictions that may be very hard for us to bear—even with God's help.
But Jesus meant something far deeper than this when He told His disciples to carry their cross. He said to them, "If anyone would come after me, he must deny himself and take up his cross daily and follow me" (Luke 9:23). In Jesus' day, a cross wasn't just a symbol of pain and suffering; it was mainly a symbol of death. What Jesus was telling them is that they needed to put to death their own plans and desires, and then turn their lives over to Him and do His will every day.
You see, Jesus doesn't simply call us to believe that He existed, or even to believe that He can save us. He calls on us to commit our whole lives to Him—to trust Him alone for our salvation, and then to follow Him as His disciples. He said, "Anyone who does not carry his cross and follow me cannot be my disciple" (Luke 14:27).
Is Christ the master of your life? Have you put to death your own plans and committed yourself to His will for your life? Don't be satisfied with anything less, for there is no greater joy in life than following Christ every day.