s1=NDSolve[{F'[x]==0.3 (1-(F[x]/0.8)Sign[1]),F[0]==-0.8},F,{x,0,5}];
s2=NDSolve[{F'[x]==0.3 (1-(F[x]/0.8)Sign[-1]),F[5]==Evaluate[F[5]/.s1]},F,{x,5,0}];
s3=NDSolve[{F'[x]==0.3 (1-(F[x]/0.8)Sign[1]),F[0]==Evaluate[F[0]/.s2]},F,{x,0,5}];
s4=NDSolve[{F'[x]==0.3 (1-(F[x]/0.8)Sign[-1]),F[5]==Evaluate[F[5]/.s3]},F,{x,5,0}];
s5=NDSolve[{F'[x]==0.3 (1-(F[x]/0.8)Sign[1]),F[0]==Evaluate[F[0]/.s4]},F,{x,0,5}];
s6=NDSolve[{F'[x]==0.3 (1-(F[x]/0.8)Sign[-1]),F[5]==Evaluate[F[5]/.s5]},F,{x,5,0}];

Plot[{Evaluate[F[x]/.s1],Evaluate[F[x]/.s2],Evaluate[F[x]/.s3],Evaluate[F[x]/. s4],Evaluate[F[x]/. s5],Evaluate[F[x]/. s6]},{x,0,5},
PlotRange->All (*, Filling->Bottom*)
]