We have the following:
Proposition Let $x: [0,T) \to \mathbb{R}$ be continuous and differentiable and satisfies $x' = f(t,x)$, then for any increasing sequence of times $T_n \nearrow T$ we have that $\limsup_{n} |x(T_n) | < \infty$.
The above proposition implies that a solution cannot "blow up in finite future time", which I think is the proper interpretation of the question asked.
The proof of the proposition is simple: intuitively using that $(x^2)' = 2xx' = 2xf$ we have that by the condition given in the hypothesis, once $|x|$ reaches a value that is larger than $R$, it can only decrease (and not increase). This forces the value of $x$ to remain bounded and hence cannot blow up in finite time. To formalise the proof we write as follows.
Proof: By way of contradiction, we assume that there exists a increasing sequence of times $T_n$ such that $|x(T_n)| > n$. Then for $n > 2R$ we have that $|x(T_{n+1})| > n+1 > 2R$. By continuity we can find the value
$$ \tau = \sup \{ t\in [T_n,T_{n+1}] | |x(t)| \leq 2R\} $$
and we know that $\tau < T_{n+1}$. Furthermore by continuity we know that $|x(\tau)| = 2R$. Apply now the mean value theorem to $|x|^2$ on the interval $[\tau,T_{n+1}]$, we have that there exists $\tau'\in [\tau,T_{n+1}]$ such that $$2x x' |_{t = \tau'} = (|x|^2)' = \frac{|x(T_{n+1}|^2 - 4R^2}{T_{n+1} - \tau} > 0 $$
On the other hand by the definition of $\tau$ we have $|x(\tau')| \geq 2R$ as $\tau' \geq \tau$. Hence we also have
$$ xx'|_{t = \tau'} = xf < 0 $$
by the hypothesis of the problem. And we arrived at the contradiction.
A comment about existence: solutions to your differential equation do not need to exist if $f$ is sufficiently bad. It is well known that a derivative of a everywhere differentiable, continuous function cannot be too discontinuous. In particular, let $\chi(x)$ denote the function that equals $1$ when $x < 0$, and $-1$ when $x \geq 0$. Now let $\phi(t)$ be a nowhere continuous function such as the indicator of the rational numbers. Then for $f(x,t) = \phi(t) \chi(x)$ it is impossible for any continuous and differentiable function $x = x(t)$ to satisfy $x' = f(t,x)$.
So strictly speaking under the conditions given it is false that every solution exists for all positive time.
