Answer
a) not possible
b) possible
Work Step by Step
a) We have $p=mv$ and $KE=\frac{1}{2}mv^2$
For $p$ to be zero, either $m=0$ or $v=0$. Either of these cases, however, will make $KE$ zero, too. Therefore, it is not possible for an object to have a kinetic energy but no momentum.
b) Consider 2 objects A and B:
$\vec{p}_A=m_A\vec{v}_A$ and $\vec{p}_B=m_B\vec{v}_B$
These 2 objects have total momentum $\sum \vec{p}=\vec{p}_A+\vec{p}_B=0$, which makes their magnitudes $p_A=p_B=p$
Now let's look at their kinetic energies:
$KE_A=\frac{1}{2}m_Av_A^2=\frac{1}{2}pv_A$ and $KE_B=\frac{1}{2}m_Bv_B^2=\frac{1}{2}pv_B$
$\sum KE=\frac{1}{2}p(v_A+v_B)\gt0$ because $p, v_A, v_B$ all are greater than 0
So it is possible to have a nonzero total kinetic energy.
