space - Is the universe like the earth?

A Universe with (hyper)spherical spatial geometry is realistic, though we don't see any evidence for it but it can't be ruled out either. As a point of interest, hyperspherical geometry doesn't necessarily mean hyperspherical topology, though it does mean the Universe must be compact i.e. of finite volume.

Now to address more thoroughly though whether it is possible to circumnavigate a spherical (in both geometric and topological senses) Universe:

Nothing can travel faster than light, so in order to find out if it is possible to circumnavigate the Universe we need to know if light travelling radially outward from some point in space will arrive at the same point in space at some later time. If $R(t_1)$ is the radius of curvature of the Universe at time $t_1$, then to circumnavigate the Universe by that time, the light must have traveled at least a proper distance of $2pi R(t_1)$ from its starting point. Or in other words the radius of the observable Universe must be at least $2pi R(t_1)$ by time $t_1$ in order for anything to circumnavigate the Universe.

The radius of the observable Universe is given by:

$r_{obs}(t_1) = R(t_1) {LARGE{int}}^{t_1}_{ t_ {int}} frac{cdt}{R(t)}$

Where $t_{int}$ is when the Universe begin (e.g. the big bang)

Therefore the condition for it to be possible to circumnavigate a spherical Universe in a rocket ship is:

${LARGE{int}}^{t_{fin}}_{ t_ {int}} frac{cdt}{R(t)} > 2pi$

Where $t_{fin}$ is the time when the Universe ends (note in a Universe without beginning $t_{int} = -infty$ and in a Universe without end $t_{fin} = infty$).

For simple models with spherical spatial geometry the value of this integral is easy to calculate:

$begin{array} {|l|l|}
hline mathbf{MODEL}&mathbf{VALUE}\
hline
mbox{Matter-dominated closed R-W}&2pi \
hline
mbox{Radiation-dominated closed R-W}&pi \
hline
mbox{De Sitter closed-slicing}&pi\
hline
mbox{Einstein static}&infty\
hline
mbox{Eddington-Lemaitre}&infty\
hline
end{array}$

This means that in the case of a Robertson-Walker closed Universe light can just circumnavigate the Universe once from the big bang to the big crunch as long as the pressure is zero. In the de Sitter closed slicing it can only make it half-way around the circumference, even though there is no big bang or big crunch. In the Einstein static Universe you can go round as many times as you like as you have an infinite amount of time to travel the circumference of fixed length. In the Eddington-Lemaitre model the expansion in early times is so slow you can circumnavigate the Universe as many times as you like.