C, 889 874 bytes

char*s="The other day,\0I met a bear,\0A great big bear,\0Oh way out there.\0He looked at me,\0I looked at him,\0He sized up me,\0I sized up him.\0He said to me,\0\"Why don't you run?\0I see you ain't,\0Got any gun.\"\0I says to him,\0\"That's a good idea.\"\0\"Now legs get going,\0get me out of here!\"\0And so I ran,\0Away from there,\0But right behind me,\0Was that bear.\0In front of me,\0There was a tree,\0A great big tree,\0Oh glory be!\0The lowest branch,\0Was ten feet up,\0So I thought I'd jump,\0And trust my luck.\0And so I jumped,\0Into the air,\0But I missed that branch,\0A way up there.\0Now don't you fret,\0And don't you frown,\0I Caught that branch,\0On the way back down!\0This is the end,\0There aint no more,\0Unless I see,\0That bear once more.\0",*p,*w;i;f(){for(w=p=s;*p;++i%4||puts(""),p=(i+4)%8?p+=strlen(p)+1:w,w=i%8?w:p)i%8>3||puts(p),puts(p);}

Compile with enabled signed integers wrap-around(-fwrapv).

This is needed because function answers must produce repeated output on subsequent calls, and the global variable i isn't reset. Since the code only cares about the remainder of division i by 4, it doesn't matter what the actual value of i is as long as it is left divisible by 4 after the function returns (and of course it is).

See it work here.

C, 889 874 bytes

char*s="The other day,\0I met a bear,\0A great big bear,\0Oh way out there.\0He looked at me,\0I looked at him,\0He sized up me,\0I sized up him.\0He said to me,\0\"Why don't you run?\0I see you ain't,\0Got any gun.\"\0I says to him,\0\"That's a good idea.\"\0\"Now legs get going,\0get me out of here!\"\0And so I ran,\0Away from there,\0But right behind me,\0Was that bear.\0In front of me,\0There was a tree,\0A great big tree,\0Oh glory be!\0The lowest branch,\0Was ten feet up,\0So I thought I'd jump,\0And trust my luck.\0And so I jumped,\0Into the air,\0But I missed that branch,\0A way up there.\0Now don't you fret,\0And don't you frown,\0I Caught that branch,\0On the way back down!\0This is the end,\0There aint no more,\0Unless I see,\0That bear once more.\0",*p,*w;i;f(){for(w=p=s;*p;++i%4||puts(""),p=(i+4)%8?p+=strlen(p)+1:w,w=i%8?w:p)i%8>3||puts(p),puts(p);}

Compile with enabled signed integers wrap-around(-fwrapv).

This is needed because function answers must produce repeated output on subsequent calls, and the function could conceivably be called repeatedly until global variable i overflows. Since the code only cares about the whether the remainder of division i by 4 and 8 is 0, it doesn't matter what the actual value of i is as long as it is left divisible by 8 after the function returns (and of course it is left divisible by 8).

See it work here.

C, 889 bytes

char*d[]={"That bear once more.","Unless I see,","There aint no more,","This is the end,","On the way back down!","I Caught that branch,","And don't you frown,","Now don't you fret,","A way up there.","But I missed that branch,","Into the air,","And so I jumped,","And trust my luck.","So I thought I'd jump,","Was ten feet up,","The lowest branch,","Oh glory be!","A great big tree,","There was a tree,","In front of me,","Was that bear.","But right behind me,","Away from there,","And so I ran,","get me out of here!\"","\"Now legs get going,","\"That's a good idea.\"","I says to him,","Got any gun.\"","I see you ain't,","\"Why don't you run?","He said to me,","I sized up him.","He sized up me,","I looked at him,","He looked at me,","Oh way out there.","A great big bear,","I met a bear,","The other day,",};t;i;f(){for(i=80;i--;i%4||puts(""))puts(d[t=i/8*4+i%4]),i%8<4||puts(d[t]);}

See it work here.

C, 889 874 bytes

char*s="The other day,\0I met a bear,\0A great big bear,\0Oh way out there.\0He looked at me,\0I looked at him,\0He sized up me,\0I sized up him.\0He said to me,\0\"Why don't you run?\0I see you ain't,\0Got any gun.\"\0I says to him,\0\"That's a good idea.\"\0\"Now legs get going,\0get me out of here!\"\0And so I ran,\0Away from there,\0But right behind me,\0Was that bear.\0In front of me,\0There was a tree,\0A great big tree,\0Oh glory be!\0The lowest branch,\0Was ten feet up,\0So I thought I'd jump,\0And trust my luck.\0And so I jumped,\0Into the air,\0But I missed that branch,\0A way up there.\0Now don't you fret,\0And don't you frown,\0I Caught that branch,\0On the way back down!\0This is the end,\0There aint no more,\0Unless I see,\0That bear once more.\0",*p,*w;i;f(){for(w=p=s;*p;++i%4||puts(""),p=(i+4)%8?p+=strlen(p)+1:w,w=i%8?w:p)i%8>3||puts(p),puts(p);}

Compile with enabled signed integers wrap-around(-fwrapv).

This is needed because function answers must produce repeated output on subsequent calls, and the global variable i isn't reset. Since the code only cares about the remainder of division i by 4, it doesn't matter what the actual value of i is as long as it is left divisible by 4 after the function returns (and of course it is).

See it work here.