Actually, a better example along the lines Otis suggests would be the geodesic $\mathbb{RP}^1\subset\mathbb{RP}^2$.  Of course,  $\mathbb{RP}^1$ is orientable and it is homologically mass-minimizing, but it can't be calibrated on any open set $U\subset\mathbb{RP}^2$ containing $\mathbb{RP}^1$ because twice it is not even stable.

Of course, this also works for any $\mathbb{RP}^{2n-1}\subset\mathbb{RP}^{2n}$, and there are higher codimension examples of closed geodesics in (orientable) lens spaces that are homologically mass-minimizing but that cannot be calibrated on any open neighborhood of the geodesic.  One can even foliate $\mathbb{RP}^3$ by homologically mass-minimizing geodesics that cannot be calibrated on any open neighborhood.

What one probably needs to assume, at least, is that every multiple of $\Sigma$ is homologically area-minimizing in some neighborhood before one could hope to construct a 'neighborhood' calibration.

**Remark (10/12/20):** I just recalled one example of possible interest for this question, since the OP is interested in what can happen in Euclidean space.  A student of mine, Timothy Murdoch, in his PhD thesis "Twisted calibrations and the cone on the Veronese surface" (Rice University, 1988), showed that the $3$-dimensional cone in $\mathbb{R}^5$ on the Veronese surface in $S^4$ is area-minimizing, but, of course, it's not orientable.  However, its 'double cover' is a cone on the $2$-sphere and so is orientable.  I don't know whether this double cover is area-minimizing in $\mathbb{R}^5$ or not.  It obviously cannot be calibrated, even if it is area-minimizing.

Explicitly, here is the example:  Think of $\mathbb{R}^5$ as $S^2_0(\mathbb{R}^3)$, the traceless $3$-by-$3$ matrices with real entries endowed with the quadratic form $\langle a,b\rangle = \mathrm{tr}(ab)$, which is invariant under $\mathrm{SO}(3)$ with the irreducible action $A\cdot a = AaA^{-1}$ for $A\in\mathrm{SO}(3)$ and $a\in S^2_0(\mathbb{R}^3)$.  Then the Veronese cone $C\subset S^2_0(\mathbb{R}^3)$ is the set of matrices $a$ with eigenvalues $t^2,t^2, -2t^2$ for some $t\ge0$.  It is a cone on an $\mathrm{SO}(3)$-homogeneous minimal surface $\mathbb{RP}^2\subset S^4$ known as the Veronese surface.  (Note that $C$ and $-C$ intersect only at the origin.)  $C$ is smooth except at the origin, and, if you define the 'double cover' by counting each smooth point as two points with different orientations, then the double cover is homeomorphic to $\mathbb{R}^3$, parametrized by the quadratic map $s:\mathbb{R}^3\to S^2_0(\mathbb{R}^3)$ defined by
$$
s(x) = |x|^2\, I_3 - 3\,x\,x^T\quad\text{for}\ x\in\mathbb{R}^3.
$$
Tim showed that, if you take the (literal) Riemannian double cover of $S^2_0(\mathbb{R}^3)\setminus (-C)$, then the double cover of $C\setminus\{0\}\simeq \mathbb{R}^3\setminus\{0\}$ can be calibrated in the ambient double cover as a Riemannian manifold.